I agree, the side veneer was removed. You can get replacement veneer at stores like Constantine’s and its fairly easy to do. The piece needs full restoration - the alligatoring was probably due to being close to a window with wild temperature swings. IMO you haven’t removed enough top coat to affect the alligatoring.

what did you use to try to rea,algamate?

When I have restored, I use Formby’s with acetone and steel wool. The Forby’s gets the top coat dissolved which you can further take off to clean stage with the Acetone. It will actually leave the stain in place. Make sure all the hardware is off because you will end up doing the whole piece. Once the piece is clean you can then add the strip to the sides to replace the veneer that is missing. I prefer using flexible veneer, contact cement with the correct roller tools so you have no bubbles. You then trim the excess off. Use a natural stain, then you can use a lacquer top coat or poly wipe for top coat. Be sure to use fine steel wool between coats with a tack cloth. If you want to get fancy, look up a French finish for the top. But you will have Popeye arms by the end of the project.

It's veneered plywood. You can sand the alligatoring smooth with 800-1500-grit abrasives, clean and re-coat until the problems disappear.

This is the before/after of my no-strip wet sanding technique. I have been doing this since 2000.

The only comment is that for a first timer, sanding can result in sanding through the veneer. Which is why I suggested the other method. It’s not an expensive piece so if you want to learn check out some books and read up on various methods. I find that I implore an arsenal of techniques and often use more than one in tackling a project. No one technique is better than another, it is simply that each piece has its own set of issues.